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Bouguer, and Maupertuis took occasion, to inves tigate by each method the nature of the meridian on different hypotheses of gravitation directed toward one centre or toward several; and they rejected all the cases, where the two methods did not agree in giving the same curve to the meridian, which frequently happened. See the Mem. of the Ac. of Paris, for 1734. But all these problems, little difficult in themselves, were in fact mere geometrical amusements. The nature of gravitation is fixed: and every other principle than that of an attraction inversely proportional to the squares of the distances is foreign to the true question.
The fundamental proposition of Newton, that the Earth is an oblate elliptical spheroid, required to be demonstrated. This was accordingly done by Stirling, in the case where, the fluid being perfectly homogencal, the oblateness is supposed to be very small. Phil. Trans. 1736 and 1737. Clairaut likewise demonstrated it on the same supposition of a very small oblateness, not only when the fluid is homogeneous, but even when it is composed of strata of different densities. In the latter case however he misled himself by considering the strata as similar; which cannot be the case when they are fluid, as he remarked himself in his Theory, of the Figure of the Earth, published in 1743.
Maclaurin was the first who demonstrated this elegant theorem, that, in 'whatever manner a homogeneous fluidnass, the particles of which attract cach other in the inverse ratio of the squares of the diştances, while at the same time it revolves round an
axis, assumed the figure of an oblate or prolate elliptical spheroid, of whatever proportions, it will remain in equilibrium, or preserve it's figure. He does not content himself with establishing the equilibrium of the central columns, both in the directions of the axes of the spheroid, and in all other directions: but he shows in addition, that any given point, taken in the interiour of the spheroid, is in equilibrium, or equally pressed upon in all directions; which is in some measure a superabundant proof. He extends this proposition to the case, where the particles of the Earth, beside their reciprocal attractions and centrifugal forces, are likewise attracted by the Sun, and by the Moon. He gives a great number of other very remarkable theorems on the attractions of ellipsoidal spheroids, which have circles or ellipses for their equator: and he applies the whole of this theory to the figure of the planets, and the phenomena of the tides. The method he adopts, to demonstrate his chief propositions, is purely synthetical; and in the opinion of geometricians is esteemed a masterpiece of invention and sagacity, equal to any thing, however admirable, that we have of Archimedes or Apollonius. See his Treatise on Fluxions, Vol. II; chap. 14.
On restricting this theory to the particular case, where the Earth, originally fluid and homogeneous, forms an oblate elliptical spheroid in consequence of attraction and the centrifugal force, we find the two axes of this spheroid to be to each other in the ratio of 230 to 229, as Newton had concluded from his suppositions, which are thereby verified.
Clairaut, who had many reasons for investigating the same question, since he had been concerned in the operations in the north, and had already demonstrated Newton's suppositions in part, composed an entire work on this subject, in which he treated the question at length, according to the laws of hydrostatics. I.
As the problems of Bouguer and Maupertuis had drawn the attention of geometricians, Clairaut deemed it incumbent on himself, to consider these likewise. He demonstrated, that under an infinite number of hypotheses of gravitation the fluid would not be in equilibrium, though the central columns reciprocally balanced each other, and the direction of gravitation were perpendicular to the surface of the fluid: he gives a general method for discriminating the hy potheses of gravitation, which admit the equilibrium, and for deterinining the figure which the fluid ought to assume, and shows, that, when the gravitation is the result of the attractions of all the parts and the centrifugal force, it is sufficient that one of the principles, either that of Huygens or that of Newton, be observed, for the other to be so likewise, and the planet to be in equilibrium. Coming then to the true state of the question, founded on the newtonian system of attraction, Clairaut first determines the figure of the Earth on the hypothesis of the homogeneity of it's particles; and here he departs from his own method to follow that of Maclaurin, which he deems preferable. Afterwards, without borrowing farther from any person, he proceeds to other very profound investigations. He explains the manner of determining
termining the variations of gravity from the equator to the pole, in a spheroid composed of strata, the densities and ellipticities of which follow any given law from the centre to the superficies: he determines the figure which the Earth would have, if, supposing it entirely fluid, it were a mass of an infinite number of fluids of different densities: he then compares his theory with observations; and in this comparison he examines the errours that must be ascribed to observations, in order that the dimensions of the terrestrial spheroid may be nearly such as the theory demands. So many new and useful views have placed this work of Clairaut among those productions of genius, which do honour to the sciences.
Still in this intricate and copious subject there re mained many important points to be elucidated, botli respecting the law of the densities of the terrestrial spheroid, and the conditions of equilibrium to which this law is subjected, according to the different cases. D'Alembert published a great number of excellent memoirs on this subject, in his Essay on the Resistance of Fluids, 1752; his Inquiries into the Mundane System, 1754; and his Mathematicul Opuscula, 1768: I regret, that they do not admit of having an abs stract of them inserted here; and must content my self with observing, that the author has given method, long wished for by geometricians, for de termining the attraction of the terrestrial spheroid on an infinite number of hypotheses beside that of the elliptical figure. He supposes, that the radius of the terrestrial spheroid is represented by an expression, which includes a constant quantity plus the series of
all the powers of the sines of latitude; and he finds the attraction, which such a spheroid exerts on a particle placed on it's surface. This includes as a particular case the common supposition, into which the square of the sine of the latitude is the only one of these powers that enters. From this important problem and it's consequences a new treatise on the figure of the Earth is formed. In this the author supposes, that the meridians are equal and similar: but by a farther effort he has likewise accomplished the determination of the attraction of a spheroid, which is not a solid of revolution; which would be of use, if the terrestrial globe had in fact an irregular figure,
Since the invention of the telescope, it has been gradually discovered, by observing the spots on their disks, that the other planets have a rotatory motion round their axes like the Earth: whence it is to be concluded, that they too are of an oblate figure; and more or less so, according as their rotation is more or less rapid. The Earth turns uniformly round it's axis in 24 hours: but on account of the inequality of it's annual elliptical orbit, and the oblique position of it's equator with respect to the ecliptic, it's days, or the intervals of time which the Sun's apparent motion occupies in returning to the same meridian, are unequal, being sometimes longer, sometimes shorter. Their mean duration is 23 hours 56 minutes. The Sun makes one revolution round it's axis, in twenty five days and half; Venus, in 23 hours, 20 minutes; Mars, in 24 hours, 40 minutes; Jupiter, in 9 hours, 56 minutes; and Saturn, in 10